Question Type 1: Modelling decay using exponential functions Exercises

Solve the equation $5^{2x-1}=125$ for $x$.

An investment of $5000 is compounded continuously at an annual rate of 4%. Calculate the value of the investment after 10 years.

Solve for $t$: $200=50\cdot2^t$.

A radioactive substance has a half-life of 6 years and an initial amount of 10 grams. Determine the decay constant $k$ and write the exponential decay model $A(t)$.

Solve for $x$ in the equation $e^{3x}=10$.

Given the model $A(t)=50e^{-0.2t}$ for the decay of a substance, find its half-life.

A sample containing 100% of a radiocarbon isotope decays with a half-life of 5730 years. What fraction of the isotope remains after 1000 years?

A population grows continuously at an annual rate of 5%. Determine the time required for the population to double.

A bacterial culture doubles in size every 3 hours. If the initial count is 100 bacteria, how long will it take to reach 10,000 bacteria?

A substance decays to 40% of its original amount in 4 days. Find the decay constant $k$ in the model $A(t)=A_0e^{kt}$, and determine how long it will take to decay to 10%.

The brightness of a star decays according to $B(t)=B_0e^{-\lambda t}$. If the brightness decreases from 1000 units to 300 units in 5 hours, find the decay constant $\lambda$ and the time required for the brightness to reach 100 units.

Solve the equation $\ln(x)+\ln(x-3)=1$ for $x$.